Mathematical weight device
In
mathematics
, a
weighing matrix
of order
and weight
is a
matrix
with entries from the set
such that:
Where
is the
transpose
of
and
is the
identity matrix
of order
. The weight
is also called the
degree
of the matrix. For convenience, a weighing matrix of order
and weight
is often denoted by
.
[3]
Weighing matrices are so called because of their use in optimally measuring the individual weights of multiple objects. When the weighing device is a
balance scale
, the
statistical variance
of the measurement can be minimized by weighing multiple objects at once, including some objects in the opposite pan of the scale where they subtract from the measurement.
[1]
[2]
Properties
[
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]
Some properties are immediate from the definition. If
is a
, then:
- The rows of
are pairwise
orthogonal
. Similarly, the columns are pairwise orthogonal.
- Each row and each column of
has exactly
non-zero elements.
- , since the definition means that
,
where
is the
inverse
of
.
- where
is the
determinant
of
.
A weighing matrix is a generalization of
Hadamard matrix
, which does not allow zero entries.
[3]
As two special cases, a
is a
Hadamard matrix
[3]
and a
is equivalent to a
conference matrix
.
Applications
[
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]
Experiment design
[
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]
Weighing matrices take their name from the problem of measuring the weight of multiple objects. If a measuring device has a statistical variance of
, then measuring the weights of
objects and subtracting the (equally imprecise)
tare weight
will result in a final measurement with a variance of
.
[4]
It is possible to increase the accuracy of the estimated weights by measuring different subsets of the objects, especially when using a
balance scale
where objects can be put on the opposite measuring pan where they subtract their weight from the measurement.
An order
matrix
can be used to represent the placement of
objects?including the tare weight?in
trials. Suppose the left pan of the balance scale adds to the measurement and the right pan subtracts from the measurement. Each element of this matrix
will have:
Let
be a column vector of the measurements of each of the
trials, let
be the errors to these measurements each
independent and identically distributed
with variance
, and let
be a column vector of the true weights of each of the
objects. Then we have:
Assuming that
is
non-singular
, we can use the
method of least-squares
to calculate an estimate of the true weights:
The variance of the estimated
vector cannot be lower than
, and will be minimum
if and only if
is a weighing matrix.
[4]
[5]
Optical measurement
[
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]
Weighing matrices appear in the
engineering
of
spectrometers
, image scanners,
[6]
and optical multiplexing systems.
[5]
The design of these instruments involve an optical mask and two detectors that measure the intensity of light. The mask can either transmit light to the first detector, absorb it, or reflect it toward the second detector. The measurement of the second detector is subtracted from the first, and so these three cases correspond to weighing matrix elements of 1, 0, and ?1 respectively. As this is essentially the same measurement problem as in the previous section, the usefulness of weighing matrices also applies.
[6]
Examples
[
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]
Note that when weighing matrices are displayed, the symbol
is used to represent ?1. Here are some examples:
This is a
:
This is a
:
This is a
:
Another
:
Which is
circulant
, i.e. each row is a
cyclic shift
of the previous row. Such a matrix is called a
and is determined by its first row.
Circulant weighing matrices are of special interest since their algebraic structure makes them easier for classification. Indeed, we know that a circulant weighing matrix of order
and weight
must be of
square
weight. So, weights
are permissible and weights
have been completely classified.
[7]
Two special (and actually, extreme) cases of circulant weighing matrices are (A) circulant Hadamard matrices which are
conjectured
not to exist unless their order is less than 5. This conjecture, the circulant Hadamard conjecture first raised by Ryser, is known to be true for many orders but is still
open
. (B)
of weight
and minimal order
exist if
is a
prime power
and such a circulant weighing matrix can be obtained by signing the complement of a
finite projective plane
.
Since all
for
have been classified, the first open case is
.
The first open case for a general weighing matrix (certainly not a circulant) is
.
Equivalence
[
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]
Two weighing matrices are considered to be
equivalent
if one can be obtained from the other by a series of permutations and negations of the rows and columns of the matrix. The classification of weighing matrices is complete for cases where
as well as all cases where
are also completed.
[8]
However, very little has been done beyond this with exception to classifying circulant weighing matrices.
[9]
[10]
Open questions
[
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]
There are many open questions about weighing matrices. The main question about weighing matrices is their existence: for which values of
and
does there exist a
? A great deal about this is unknown. An equally important but often overlooked question about weighing matrices is their enumeration: for a given
and
, how many
's are there?
This question has two different meanings. Enumerating up to equivalence and enumerating different matrices with same
n
,
k
parameters. Some papers were published on the first question but none were published on the second important question.
References
[
edit
]
- ^
a
b
Raghavarao, Damaraju (1960).
"Some Aspects of Weighing Designs"
.
The Annals of Mathematical Statistics
.
31
(4). Institute of Mathematical Statistics: 878?884.
doi
:
10.1214/aoms/1177705664
.
ISSN
0003-4851
.
- ^
a
b
Seberry, Jennifer (2017). "Some Algebraic and Combinatorial Non-existence Results".
Orthogonal Designs
. Cham: Springer International Publishing. pp. 7?17.
doi
:
10.1007/978-3-319-59032-5_2
.
ISBN
978-3-319-59031-8
.
- ^
a
b
c
Geramita, Anthony V.; Pullman, Norman J.; Wallis, Jennifer S. (1974).
"Families of weighing matrices"
.
Bulletin of the Australian Mathematical Society
.
10
(1). Cambridge University Press (CUP): 119?122.
doi
:
10.1017/s0004972700040703
.
ISSN
0004-9727
.
S2CID
122560830
.
- ^
a
b
Raghavarao, Damaraju (1971). "Weighing Designs".
Constructions and combinatorial problems in design of experiments
. New York: Wiley. pp. 305?308.
ISBN
978-0471704850
.
- ^
a
b
Koukouvinos, Christos; Seberry, Jennifer (1997).
"Weighing matrices and their applications"
.
Journal of Statistical Planning and Inference
.
62
(1). Elsevier BV: 91?101.
doi
:
10.1016/s0378-3758(96)00172-3
.
ISSN
0378-3758
.
S2CID
122205953
.
- ^
a
b
c
Sloane, Neil J. A.; Harwit, Martin (1976-01-01). "Masks for Hadamard transform optics, and weighing designs".
Applied Optics
.
15
(1). The Optical Society: 107?114.
Bibcode
:
1976ApOpt..15..107S
.
doi
:
10.1364/ao.15.000107
.
ISSN
0003-6935
.
PMID
20155192
.
- ^
Arasu, K.T.; Gordon, Daniel M.; Zhang, Yiran (2019). "New Nonexistence Results on Circulant Weighing Matrices".
arXiv
:
1908.08447v3
.
- ^
Harada, Masaaki; Munemasa, Akihiro (2012). "On the classification of weighing matrices and self-orthogonal codes".
J. Combin. Designs
.
20
: 40?57.
arXiv
:
1011.5382
.
doi
:
10.1002/jcd.20295
.
S2CID
1004492
.
- ^
Ang, Miin Huey; Arasu, K.T.; Lun Ma, Siu; Strassler, Yoseph (2008).
"Study of proper circulant weighing matrices with weight 9"
.
Discrete Mathematics
.
308
(13): 2802?2809.
doi
:
10.1016/j.disc.2004.12.029
.
- ^
Arasu, K.T.; Hin Leung, Ka; Lun Ma, Siu; Nabavi, Ali; Ray-Chaudhuri, D.K. (2006).
"Determination of all possible orders of weight 16 circulant weighing matrices"
.
Finite Fields and Their Applications
.
12
(4): 498?538.
doi
:
10.1016/j.ffa.2005.06.009
.